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Theorem cador 1544
Description: The adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
Assertion
Ref Expression
cador (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))

Proof of Theorem cador
StepHypRef Expression
1 xor2 1467 . . . . . . 7 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
21rbaib 946 . . . . . 6 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ (𝜑𝜓)))
32anbi1d 740 . . . . 5 (¬ (𝜑𝜓) → (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ 𝜒)))
4 ancom 466 . . . . 5 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜒 ∧ (𝜑𝜓)))
5 andir 911 . . . . 5 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
63, 4, 53bitr3g 302 . . . 4 (¬ (𝜑𝜓) → ((𝜒 ∧ (𝜑𝜓)) ↔ ((𝜑𝜒) ∨ (𝜓𝜒))))
76pm5.74i 260 . . 3 ((¬ (𝜑𝜓) → (𝜒 ∧ (𝜑𝜓))) ↔ (¬ (𝜑𝜓) → ((𝜑𝜒) ∨ (𝜓𝜒))))
8 df-or 385 . . 3 (((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))) ↔ (¬ (𝜑𝜓) → (𝜒 ∧ (𝜑𝜓))))
9 df-or 385 . . 3 (((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))) ↔ (¬ (𝜑𝜓) → ((𝜑𝜒) ∨ (𝜓𝜒))))
107, 8, 93bitr4i 292 . 2 (((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))))
11 df-cad 1543 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))
12 3orass 1039 . 2 (((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ ((𝜑𝜒) ∨ (𝜓𝜒))))
1310, 11, 123bitr4i 292 1 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  wxo 1461  caddwcad 1542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-xor 1462  df-cad 1543
This theorem is referenced by:  cadan  1545  cadnot  1551
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